US8346711B2  Method for identifying multiinput multioutput Hammerstein models  Google Patents
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Abstract
Description
1. Field of the Invention
The present invention relates to Hammerstein models, and particularly to a computerized method for identifying multiinput multioutput (MIMO) Hammerstein models for engineering design applications.
2. Description of the Related Art
The Hammerstein Model belongs to a family of block oriented models, and is made up of a memoryless nonlinear part followed by a linear dynamic part. It has been known to effectively represent and approximate several industrial processes, such as, for example, pH neutralization processes, distillation column processes, and heat exchange processes. Hammerstein models have also been used to successfully model nonlinear filters, biological systems, water heaters, and electrical drives.
A significant amount of research has been carried out on identification of Hammerstein models. Systems can be modeled by employing either nonparametric or parametric models. Nonparametric representations involve kernel regression or expansion of series, such as the Volterra series. This results in a theoretically infinite number of model parameters, and is therefore represented in terms of curves, such as step responses or bode diagrams. Parametric representations, such as statespace models, are more compact, as they have fewer parameters and the nonlinearity is expressed as a linear combination of finite and known functions.
Development of nonlinear models is the critical step in the application of nonlinear model based control strategies. Nonlinear behavior is prominent in the dynamic behavior of physical systems. Most physical devices have nonlinear characteristics outside a limited linear range. In most chemical processes, for example, understanding the nonlinear characteristics is important for designing controllers that regulate the process. It is rather difficult, yet necessary, to select a reasonable structure for the nonlinear model to capture the process nonlinearities. The nonlinear model used for control purposes should be as simple as possible, warranting minimal computational load and, at the same time, retaining most of the nonlinear dynamic characteristics of the system. The following convention has been used in what follows: upper case variables in bold represent matrices, lower case bold variables represent vectors, and lower case regular (i.e., nonbold) variables represent scalar quantities.
Many model structures have been proposed for the identification of nonlinear systems. The nonlinear static block followed by a dynamic block in the Hammerstein structure has been found to be a simple and effective representation for capturing the dynamics of typical chemical engineering processes such as distillation columns and heat exchangers, for example. Nonlinear system identification involves the following tasks: Structure selection, including selection of suitable nonlinear model structures and the number of model parameters; input sequence design, including the determination of the input sequence u(t) which is injected into the system to generate the output sequence y(t); noise modeling, which includes the determination of the dynamic model which generates the noise input w(t); parameter estimation, which includes estimation of the remaining model parameters from the dynamic system data u(t) and y(t), and the noise input w(t); and model validation, including the comparison of system data and model predictions for data not used in model development.
Hammerstein systems can be modeled by employing either nonparametric or parametric models. Nonparametric models represent the system in terms of curves resulting from expansion of series, such as the Volterra series or kernel regression. In practice, these curves are sampled, often leading to a large number of parameters. Parametric representations, such as statespace models, are more compact and have fewer parameters, while the nonlinearity is expressed as a linear combination of finite and known functions. In parametric identification, the Hammerstein model is represented by the following set of equations:
y(t)=−a _{1} y(t−1)− . . . −a _{n} y(t−n)+b _{o} v(t)+ . . . +b _{m} v(t−m) (1)
v(t)=c _{1} u(t)+c _{2} u ^{2}(t)+ . . . +c _{L} u ^{L}(t) (2)
where v(t) describes the nonlinearity, L is the order of the nonlinearity, and y(t) and u(t) are the outputs and inputs of the system.
In MIMO Hammerstein models, as noted above, a nonlinear system is represented as a nonlinear memoryless subsystem f(.), followed by a linear dynamic part. The input sequence u(t) and the output sequence y(t) are accessible to measurements, but the intermediate signal sequence v(t) is not. As shown in
Many different techniques have been proposed for the blackbox estimation of Hammerstein systems from inputoutput measurements. These techniques mainly differ in the way that static nonlinearity is represented and in the type of optimization problem that is finally obtained. In parametric approaches, the static nonlinearity is expressed in a finite number of parameters. Both iterative and noniterative methods have been used for determination of the parameters of the staticnonlinear and lineardynamic parts of the model. Typical techniques, however, are extremely costly in terms of computational time and energy.
Additionally, most techniques designed to deal with Hammerstein models focus purely on singleinput singleoutput (SISO) models. Identification of MIMO systems, however, is a problem which has not been well explored. Identification based on prediction error methods (PEM), for example, is a complicated function of the system parameters, and has to be solved by iterative descent methods, which may get stuck into local minima. Further, optimization methods need an initial estimate for a canonical parametrization model; i.e. models with minimal numbers of parameters, which might not be easy to provide.
It has been shown that this minimal parametrization can lead to several problems. PEM have, therefore, inherent difficulties with MIMO system identification. More recent studies have also shown that maximum likelihood criterion results in a nonconvex optimization problem in which global optimization is not guaranteed. Subspace identification methods (SIM) do not need nonlinear optimization techniques, nor do these methods need to impose to the system a canonical form. Subspace methods therefore do not suffer from the inconveniences encountered in applying PEM methods to MIMO system identification. Thus, it would be desirable to make use of this advantage, modeling the linear dynamic subsystem of the Hammerstein model with a statespace model rather than polynomial models. Thus, a method for identifying multiinput multioutput Hammerstein models solving the aforementioned problems is desired.
The present invention relates to the identification of multiinput multioutput (MIMO) Hammerstein models and, in particular, to a method utilizing a particle swarm optimization (PSO) subspace algorithm. The inventive method includes modeling of the linear dynamic part of a Hammerstein model with a statespace model, and modeling the nonlinear part of the Hammerstein model with a radial basis function neural network (RBFNN). Accurate identification of a Hammerstein model requires that output error between the actual and estimated systems be minimized, thus the problem of identification is, in essence, an optimization problem. The PSOsubspace algorithm of the present method is an optimization algorithm. Particle swarm optimization (PSO), typically known for its heuristic search capabilities, is used for estimating the parameters of the RBFNN. Parameters of the linear part are estimated using a numerical algorithm for subspace statespace system identification (N4SID).
The present method includes the following steps: (a) estimating an initial set of state space matrices A, B, C, and D using nonlinear data acquired from the plant (these matrices are estimated using the methods of subspace identification); (b) initializing a swarm of particles with a random population of possible radial basis function neural network weights; (c) initializing a particle swarm optimization method with a random population of possible radial basis function neural network weights; (d) calculating a global best set of weights which minimizes an output error measure; (e) estimating sets of radial basis function neural network outputs v(t) for all values of t based upon the global best set of weights; (f) estimating the statespace matrices A, B, C, and D from the radial basis function neural network outputs estimated in step (e) and a set of original system outputs y(t) for values of t; (g) calculating a set of system outputs ŷ(t) from the estimated statespace matrices A, B, C, and D in step (f); (h) calculating the output error measure; and (i) repeating steps (c) to (h) if the calculated output error measure is greater than a preselected threshhold error measure.
These and other features of the present invention will become readily apparent upon further review of the following specification and drawings.
Similar reference characters denote corresponding features consistently throughout the attached drawings.
The present invention relates to the identification of MIMO Hammerstein models and, in particular, to a method utilizing a particle swarm optimization (PSO) subspace algorithm. The method includes modeling of the linear dynamic part of a Hammerstein model with a statespace model, and modeling the nonlinear part of the Hammerstein model with a radial basis function neural network (RBFNN). Accurate identification of a Hammerstein model requires that output error between the actual and estimated systems be minimized, thus the problem of identification is, in essence, an optimization problem. The PSOsubspace algorithm of the present method is an optimization algorithm. Particle swarm optimization (PSO), typically known for its heuristic search capabilities, is used for estimating the parameters of the RBFNN. Parameters of the linear part are estimated using a numerical algorithm for subspace statespace system identification (N4SID).
As noted above, the linear dynamic part of the Hammerstein model is modeled by a statespace model. The static nonlinear part of the Hammerstein model is modeled using RBFNN. The external inputs to the system u(t) are fed to the RBFNN, as best illustrated in
Particularly, considering a MIMO Hammerstein system with p inputs and r outputs, the input layer of the RBFNN takes the system inputs u(t)=[u_{1}(t) . . . u_{p}(t)]^{T }and the second layer of the RBFNN, referred to as the “hidden layer”, performs a fixed nonlinear transformation on this data using the basis function φ. The output layer of the RBFNN then implements a linear weightage and supplies the response of the network to the output v(t)=[v_{1}(t) . . . v_{p}(t)]^{T}. Thus, the only adjustable parameters in the RBFNN are the weights of its output layer. This output, v(t) is fed to the linear subsystem whose output is given by:
x(t+1)=Ax(t)+Bv(t)+s(t) (3a)
ŷ(t)=Cx(t)+Dv(t)+z(t) (3b)
where v(t)ε
where E denotes expected value and δ_{pq }denotes a Kronecker delta function.
Accurate identification of a Hammerstein model requires that the error between the outputs of the actual and estimated systems be minimized. If y(t)=[y_{1}(t)]^{T }denotes the vector for original outputs of the sampled data, and ŷ(t)=[ŷ_{1}(t) . . . ŷ_{r}(t)]^{T }denotes the vector for the outputs of the estimated system, then a cost function based on the square of the output error is sought to be minimized, with the cost function being given by:
where N denotes the number of data points, e(t)=[e_{1}(t) . . . e_{r}(t)]^{T }is the vector for output error at discrete time instant t, and is given by e(t)=Y(t)−Ŷ(t).
The PSO plays a large part in training the RBFNN. In a swarm of particles, where each particle represents a candidate value for the weight of RBFNN output layer, the fitness of the particles is the reciprocal of the cost index given in equation (5). Hence, the smaller the sum of output errors, the more fit are the particles. Based on this principle, the PSO updates the position of all the particles moving towards an optimal solution for the weights of RBFNN.
Hammerstein identification is, therefore, solved as an optimization problem in which PSO is used to estimate the parameters of RBFNN, while parameters of the linear subsystem are estimated using the N4SID numerical subspace algorithm. RBFNN is an effective type of neural network that has proved useful in applications such as function approximation and pattern recognition. It should be noted that the static nonlinearity in a MIMO Hammerstein model can be either combined or separate.
where Q is the number of neurons in the hidden layer, c_{i }is the center for the i^{th }neuron of that layer, w_{i }is the weight connecting the i^{th }neuron node to the output layer, φ is the radial basis function, and ∥.∥ denotes the norm.
In the second system of
where u(t)ε
Subspace identification is used for estimating the parameters of the linear dynamic part of the model; i.e., the matrices of the statespace model. The present inventive method makes use of the N4SID numerical algorithm. The objective of the algorithm is to determine the order n of the system, and the system matrices Aε
^{n×n}, Bε ^{n×P}, Cε ^{R×n}, D ε ^{R×P}, Qε ^{n×n}, Rε ^{R×R}, and Sε ^{n×R }(and the Kalman gain matrix K if required), without any prior knowledge of the structure of the system. This is achieved in two main steps: First, model order n and a Kalman filter state sequence of estimates {circumflex over (x)}_{i},{circumflex over (x)}_{i+1}, . . . ,{circumflex over (x)}_{i+j }are determined by projecting row spaces of data block Hankel matrices, and then applying a singular value decomposition. Next, the solution of a least squares problem is used to obtain the state space matrices A, B, C, and D.PSO is a heuristic searchbased optimzation technique which exhibits behavior of swarm intelligence. PSO differs from other evolutionary algorithms (EAs) in that it changes its population from one iteration to the next. Unlike genetic algorithms (GAs) and other EAs, operators like selection, mutation and crossover are not used to change the population. Existing particles are, instead, modified according to a pair of formulae. PSO thus differs from other EAs in terms of performance. PSO is more robust and faster in comparison, and also provides solutions to complex and nonlinear problems, with relative ease in finding a global minimum. The convergence characteristic of PSO is relatively stable and provides solutions in relatively little time.
PSO begins with a population of particles. Each particle adjusts its position according to its own experience, as well as by the experience of neighboring particles. Each particle is treated as a point in Ddimensional space. The i^{th }particle is represented as:
X _{i}=(x _{i1} ,x _{i2} , . . . ,x _{iD}) (8)
and the best positions of the particles (the position giving the most optimum solution) are recorded as:
P _{i}=(p _{i1} ,p _{i2} , . . . ,p _{iD}) (9)
with the change in position (velocity) of each particle being given as:
V _{i}=(v _{i1} ,v _{i2} , . . . ,v _{iD}) (10)
where the velocity and positions of the particles are updated according to the following pair of equations:
V _{i} ^{n+1} =V _{i} ^{n} +c _{1} *r _{i1} ^{n}*(P _{i} ^{n} −X _{i} ^{n})+c _{2} *r _{i2} ^{n}*(P _{g} ^{n} −X _{i} ^{n}) (11a)
X _{i} ^{n+1} =X _{i} ^{n} +x*V _{i} ^{n+1} (11b)
where c_{1 }and c_{2 }are two positive real constants, the cognitive parameter and the social parameter, respectively. The value of c_{1 }signifies how much a particle trusts its past experiences, and how much it searches for solutions around the local best position, while the value of c_{2 }determines the swarm's attraction toward a global best position.
Higher values of c_{1 }and c_{2 }make the swarm able to react faster to changes, whereas lower values of these parameters make the swarm slower in changing neighborhoods of solutions. The present inventive method makes use of values such that c_{1}≧c_{2}, and c_{1}+c_{2}≦4.
Introducing an inertia weight w and a constriction factor X, the update equations become:
V _{i} ^{n+1} =X(wV _{i} ^{n} +c _{1} *r _{i1} ^{n}*(P _{i} ^{n} −X _{i} ^{n})+c _{2} *r _{i2} ^{n}*(P _{g} ^{n} −X _{i} ^{n})) (12a)
X _{i} ^{n+1} =X _{i} ^{n} +x*V _{i} ^{n+1} (12b)
where w is the inertial weight, and X is the constriction factor, which is used to limit the velocity and help better convergence. The value of c_{1 }signifies how much a particle trusts its past experiences, and how much it is attracted to a local best position while the value of c_{2 }determines the swarm's attraction towards a global best position.
The method relates to Hammerstein model identification, which, in essence, can be summarized as follows: Given a set of N noisy inputs u(t)_{t=0} ^{N−1 }and outputs y(t)_{t=0} ^{N−1}, find the weights of the RBFNN; and find the matrices of the state space model. Since the output y(t) is nonlinear in relation to the input u(t), the calculations are nontrivial. Thus, a recursive algorithm is utilized to update the weights of the neural network for each set of input and output data.
As illustrated in
In a first example, a two input, two output Hammerstein type nonlinear process with two separate static nonlinearities is considered. The first nonlinearity is a saturation nonlinearity, while the second nonlinearity is a tan h/exponential function:
with the dynamic linear part being given by the following secondorder discrete time statespace system:
where the linear part of the system has eigen values at λ_{1.2}=0.5±0.5i.
The system is identified using a PSO/subspace algorithm. A set of persistently exciting data is generated in the intervals [−1,1] and [0,4] for the two inputs, respectively. The centers of the RBFNNs are initialized with a set of evenly distributed centers in their respective intervals. The PSO/subspace algorithm identifies the system within a few iterations of the algorithm and the mean squared error converges to a final value of 9×10^{−4}.
In a second example of the method, the Hammerstein type nonlinear process includes a static nonlinearity given by:
with the dynamic linear part being given by a thirdorder discrete time statespace system:
where, for this example, an RBF network of ten neurons is initialized with centers evenly distributed in the interval [−1.75,1.75]. The linear part of the system has eigen values at λ_{1}=0.7, λ_{2}=0.6, and λ_{3}=0.5.
Persistently exciting data is generated in the intervals [−1.75,1.75] and [−4,4] for the two inputs, respectively. The centers of the RBFNNs are initialized evenly distributed within the input intervals. The PSO/subspace method identifies the system with a good estimate of the nonlinearity shape. The mean squared error converges to a final value of 7×10^{−4}.
As seen from the plots, the shapes of the nonlinearities have been estimated quite accurately. The method combines the advantages of PSO with those of statespace models. PSO is well know to outperform other EAs in finding global optima, and statespace models can more easily be extended to systems with multiple inputs and outputs, as compared to polynomial models.
In the above, the calculations may be performed by any suitable computer system, such as that diagrammatically shown in
Processor 114 may be associated with, or incorporated into, any suitable type of computing device, for example, a personal computer or a programmable logic controller. The display 118, the processor 114, the memory 112 and any associated computer readable recording media and/or communication transmission media are in communication with one another by any suitable type of data bus, as is well known in the art.
Examples of computerreadable recording media include a magnetic recording apparatus, an optical disk, a magnetooptical disk, and/or a semiconductor memory (for example, RAM, ROM, etc.). Examples of magnetic recording apparatus, which may be used in addition to memory 112, or in place of memory 112, include a hard disk device (HDD), a flexible disk (FD), and a magnetic tape (MT). Examples of the optical disk include a DVD (Digital Versatile Disc), a DVDRAM, a CDROM (Compact DiscRead Only Memory), and a CDR (Recordable)/RW.
It is to be understood that the present invention is not limited to the embodiments described above, but encompasses any and all embodiments within the scope of the following claims.
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